{"trustable":true,"prependHtml":"\u003cstyle type\u003d\u0027text/css\u0027\u003e\n .input, .output {\n border: 1px solid #888888;\n }\n .output {\n margin-bottom: 1em;\n position: relative;\n top: -1px;\n }\n .output pre, .input pre {\n background-color: #EFEFEF;\n line-height: 1.25em;\n margin: 0;\n padding: 0.25em;\n }\n \u003c/style\u003e\n \u003clink rel\u003d\"stylesheet\" href\u003d\"//codeforces.org/s/96598/css/problem-statement.css\" type\u003d\"text/css\" /\u003e\u003cscript\u003e window.katexOptions \u003d { disable: true }; \u003c/script\u003e\n\u003cscript type\u003d\"text/x-mathjax-config\"\u003e\n MathJax.Hub.Config({\n tex2jax: {\n inlineMath: [[\u0027$$$\u0027,\u0027$$$\u0027], [\u0027$\u0027,\u0027$\u0027]],\n displayMath: [[\u0027$$$$$$\u0027,\u0027$$$$$$\u0027], [\u0027$$\u0027,\u0027$$\u0027]]\n }\n });\n\u003c/script\u003e\n\u003cscript type\u003d\"text/javascript\" async src\u003d\"https://mathjax.codeforces.org/MathJax.js?config\u003dTeX-AMS_HTML-full\"\u003e\u003c/script\u003e","sections":[{"title":"","value":{"format":"HTML","content":"\u003cp\u003e\u003cspan class\u003d\"tex-font-style-it\"\u003e— This is not playing but duty as allies of justice, Nii-chan!\u003c/span\u003e\u003c/p\u003e\u003cp\u003e\u003cspan class\u003d\"tex-font-style-it\"\u003e— Not allies but justice itself, Onii-chan!\u003c/span\u003e\u003c/p\u003e\u003cp\u003eWith hands joined, go everywhere at a speed faster than our thoughts! This time, the Fire Sisters\u0026nbsp;— Karen and Tsukihi\u0026nbsp;— is heading for somewhere they\u0027ve never reached\u0026nbsp;— water-surrounded islands!\u003c/p\u003e\u003cp\u003eThere are three clusters of islands, conveniently coloured red, blue and purple. The clusters consist of \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003ea\u003c/i\u003e\u003c/span\u003e, \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003eb\u003c/i\u003e\u003c/span\u003e and \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003ec\u003c/i\u003e\u003c/span\u003e distinct islands respectively.\u003c/p\u003e\u003cp\u003eBridges have been built between some (possibly all or none) of the islands. A bridge bidirectionally connects two different islands and has length \u003cspan class\u003d\"tex-span\"\u003e1\u003c/span\u003e. For any two islands of the same colour, either they shouldn\u0027t be reached from each other through bridges, or the shortest distance between them is \u003cspan class\u003d\"tex-font-style-bf\"\u003eat least \u003cspan class\u003d\"tex-span\"\u003e3\u003c/span\u003e\u003c/span\u003e, apparently in order to prevent oddities from spreading quickly inside a cluster.\u003c/p\u003e\u003cp\u003eThe Fire Sisters are ready for the unknown, but they\u0027d also like to test your courage. And you\u0027re here to figure out the number of different ways to build all bridges under the constraints, and give the answer modulo \u003cspan class\u003d\"tex-span\"\u003e998 244 353\u003c/span\u003e. Two ways are considered different if a pair of islands exist, such that there\u0027s a bridge between them in one of them, but not in the other.\u003c/p\u003e"}},{"title":"Input","value":{"format":"HTML","content":"\u003cp\u003eThe first and only line of input contains three space-separated integers \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003ea\u003c/i\u003e\u003c/span\u003e, \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003eb\u003c/i\u003e\u003c/span\u003e and \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003ec\u003c/i\u003e\u003c/span\u003e (\u003cspan class\u003d\"tex-span\"\u003e1 ≤ \u003ci\u003ea\u003c/i\u003e, \u003ci\u003eb\u003c/i\u003e, \u003ci\u003ec\u003c/i\u003e ≤ 5 000\u003c/span\u003e)\u0026nbsp;— the number of islands in the red, blue and purple clusters, respectively.\u003c/p\u003e"}},{"title":"Output","value":{"format":"HTML","content":"\u003cp\u003eOutput one line containing an integer\u0026nbsp;— the number of different ways to build bridges, modulo \u003cspan class\u003d\"tex-span\"\u003e998 244 353\u003c/span\u003e.\u003c/p\u003e"}},{"title":"Examples","value":{"format":"HTML","content":"\u003ctable class\u003d\u0027vjudge_sample\u0027\u003e\n\u003cthead\u003e\n \u003ctr\u003e\n \u003cth\u003eInput\u003c/th\u003e\n \u003cth\u003eOutput\u003c/th\u003e\n \u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd\u003e\u003cpre\u003e1 1 1\n\u003c/pre\u003e\u003c/td\u003e\n \u003ctd\u003e\u003cpre\u003e8\n\u003c/pre\u003e\u003c/td\u003e\n \u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n"}},{"title":"","value":{"format":"HTML","content":"\u003ctable class\u003d\u0027vjudge_sample\u0027\u003e\n\u003cthead\u003e\n \u003ctr\u003e\n \u003cth\u003eInput\u003c/th\u003e\n \u003cth\u003eOutput\u003c/th\u003e\n \u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd\u003e\u003cpre\u003e1 2 2\n\u003c/pre\u003e\u003c/td\u003e\n \u003ctd\u003e\u003cpre\u003e63\n\u003c/pre\u003e\u003c/td\u003e\n \u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n"}},{"title":"","value":{"format":"HTML","content":"\u003ctable class\u003d\u0027vjudge_sample\u0027\u003e\n\u003cthead\u003e\n \u003ctr\u003e\n \u003cth\u003eInput\u003c/th\u003e\n \u003cth\u003eOutput\u003c/th\u003e\n \u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd\u003e\u003cpre\u003e1 3 5\n\u003c/pre\u003e\u003c/td\u003e\n \u003ctd\u003e\u003cpre\u003e3264\n\u003c/pre\u003e\u003c/td\u003e\n \u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n"}},{"title":"","value":{"format":"HTML","content":"\u003ctable class\u003d\u0027vjudge_sample\u0027\u003e\n\u003cthead\u003e\n \u003ctr\u003e\n \u003cth\u003eInput\u003c/th\u003e\n \u003cth\u003eOutput\u003c/th\u003e\n \u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd\u003e\u003cpre\u003e6 2 9\n\u003c/pre\u003e\u003c/td\u003e\n \u003ctd\u003e\u003cpre\u003e813023575\n\u003c/pre\u003e\u003c/td\u003e\n \u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n"}},{"title":"Note","value":{"format":"HTML","content":"\u003cp\u003eIn the first example, there are \u003cspan class\u003d\"tex-span\"\u003e3\u003c/span\u003e bridges that can possibly be built, and no setup of bridges violates the restrictions. Thus the answer is \u003cspan class\u003d\"tex-span\"\u003e2\u003csup class\u003d\"upper-index\"\u003e3\u003c/sup\u003e \u003d 8\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eIn the second example, the upper two structures in the figure below are instances of valid ones, while the lower two are invalid due to the blue and purple clusters, respectively.\u003c/p\u003e\u003ccenter\u003e \u003cimg class\u003d\"tex-graphics\" src\u003d\"CDN_BASE_URL/5c0f5c98de8d0341a4b3a8c2f13493f4?v\u003d1713703491\" style\u003d\"max-width: 100.0%;max-height: 100.0%;\"\u003e \u003c/center\u003e"}}]}